Orbit sizes of p-groups
type: Item
instance of: scholarly article
title: Orbit sizes of p-groups
zbMATH Open document ID: 0699.20006
DOI: 10.1007/BF01198103
author: Bertram Huppert
author: Olaf Manz
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: When does a p-group P, acting faithfully and coprimely on a solvable group H, have a long orbit (preferably a regular orbit) in this action? Around this question [see \textit{D. Passman}, Trans. Am. Math. Soc. 123, 99-111 (1966; Zbl 0139.018), or \textit{T. Berger}, J. Algebra 51, 416-424 (1978; Zbl 0377.20007), or \textit{P. Fleischmann}, J. Algebra 103, 211-215 (1986; Zbl 0604.20014), for instance] several authors have found results. Here some general and easy ideas are proposed, relying on a paper of \textit{R. Gow} [J. Algebra 65, 421-426 (1980; Zbl 0445.20003)]. There are four lemmas and three theorems in the paper under review, not new, but proved by means of the ideas mentioned before. We state two of the theorems. Theorem 4. Let the non-trivial finite p- group P act irreducibly and faithfully on the finite dimensional vector space V over \({\mathbb{F}}_ q\), q prime, \(q\neq p\). Then P induces at least two regular orbits on V, whenever the ordered pair \(\{\) p,q\(\}\) does not belong to the set \(\{\) \(\{\) 2, Mersenne prime\(\}\), \(\{\) 2, Fermat prime\(\}\), \(\{\) Fermat prime, \(2\}\) \(\}\). Theorem 8. Suppose that the finite p-group P contains no section isomorphic to \(C_ p wr C_ p\). Then P has a regular orbit in its action on any finite dimensional vector space V over \({\mathbb{F}}_ q\), where \(p\neq q\), q prime.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 4146230
zbMATH Keywords: action
zbMATH Keywords: finite p-group
zbMATH Keywords: finite dimensional vector space
zbMATH Keywords: regular orbits
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
cites work: Hall-Higman type theorems. VI
cites work: Finite groups with regular orbits on vector spaces
cites work: On the number of characters in a p-block of a p-solvable group
cites work: Q4200361
cites work: Q3934684
cites work: Groups with Normal, Solvable Hall p � -Subgroups
full work available at URL: https://doi.org/10.1007/bf01198103
OpenAlex ID: W2078293995
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description: scientific article; zbMATH DE number 4146230